The theory of the fluctuations in brightness of the Milky way. I.

Abstract

In this paper the following statistical problem is considered. Let stars and interstellar clouds occur with a uniform distribution. Let the system extend to a linear distance L in the direction of a line of sight. Let a cloud reduce the intensity of the light of the stars immediately behind it by a factor q. Let the occurrence of clouds with a transparency factor q be governed by a frequency function ψ(q). Given all this, - it is required to find the probability distribution, g(I, L), of the observed brightness, I. From a consideration of this problem it is shown that the following integral equation governs the distribution of brightness: g(u+ξ)+∂g/∂u+∂g/∂ξ=∫¹₀g(u/q,ξ)ψ(q)dq/q, where u is the observed brightness measured in suitable units and ξ is the average number of clouds in the direction of the line of sight. It is further shown that the foregoing integral equation enables us to obtain explicit formulae for all the moments of g as functions of ξ and the moments of ψ(q). As an illustration of the use of these general formulae for the moments, an example investigated by Markarian has been reconsidered in an attempt to derive the mean and mean-square deviation of the optical thicknesses of the interstellar clouds.